We present numerical cases to validate the a posteriori error estimate of Theorem REF  and show its use in the framework of an adaptive algorithm.
The simulations are performed with the open source finite element library FreeFem++ (see [1]} and also https://freefem.org/).
We will use the notion of local and global total estimator: \(\eta _{\rm tot,T}\)  defined by (REF ) and
\(\eta _{\text{tot}} \left(\sum _{T\in \mathcal {T}_h} \left(\eta _{\text{tot},T}\right)^2\right)^{{1}{2}}.\) 
